Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{p^2 - 81}{p + 9}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = p$ $ b = \sqrt{81} = 9$ So we can rewrite the expression as: $r = \dfrac{({p} + {9})({p} {-9})} {p + 9} $ We can divide the numerator and denominator by $(p + 9)$ on condition that $p \neq -9$ Therefore $r = p - 9; p \neq -9$